Hostname: page-component-586b7cd67f-g8jcs Total loading time: 0 Render date: 2024-11-26T08:22:37.307Z Has data issue: false hasContentIssue false
  • English
  • Français

An Inductive Julia-Carathéodory Theorem for Pick Functions in Two Variables

Part of: Holomorphic functions of several complex variables Linear function spaces and their duals

Published online by Cambridge University Press:  10 April 2018

J. E. Pascoe
J. E. Pascoe*
Affiliation:
Washington University, St. Louis, MO 63130, USA ([email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

Classically, Nevanlinna showed that functions from the complex upper half plane into itself which satisfy nice asymptotic conditions are parametrized by finite measures on the real line. Furthermore, the higher order asymptotic behaviour at infinity of a map from the complex upper half plane into itself is governed by the existence of moments of its representing measure, which was the key to his solution of the Hamburger moment problem. Agler and McCarthy showed that an analogue of the above correspondence holds between a Pick function f of two variables, an analytic function which maps the product of two upper half planes into the upper half plane, and moment-like quantities arising from an operator theoretic representation for f. We apply their ‘moment’ theory to show that there is a fine hierarchy of levels of regularity at infinity for Pick functions in two variables, given by the Löwner classes and intermediate Löwner classes of order N, which can be exhibited in terms of certain formulae akin to the Julia quotient.

Keywords

higher order Julia-Carathéodory theoremasymptotic boundary behaviourHankel vector moment sequencestwo complex variablesfunction theory on the bidisk

MSC classification

Primary: 32A70: Functional analysis techniques
Secondary: 46E22: Hilbert spaces with reproducing kernels (=
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 61 , Issue 3 , August 2018 , pp. 647 - 660
Copyright
Copyright © Edinburgh Mathematical Society 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Abate, M., The Julia-Wolff-Carathéodory theorem in polydisks, J. Anal. Math. 74 (1998), 275–306.Google Scholar
2.Abate, M., Angular derivatives in several complex variables, In Real methods in complex and CR geometry, (ed. Zaitsev, D. and Zampieri, G.), Lecture Notes in Mathematics, Volume 1848, pp. 1–47 (Springer, Berlin, 2004).CrossRefGoogle Scholar
3.Agler, J. and McCarthy, J. E., Hankel vector moment sequences and the non-tangential regularity at infinity of two variable pick functions, Trans. Amer. Math. Soc. 368(5) (2014), 1379–1411.Google Scholar
4.Agler, J., McCarthy, J. E. and Young, N. J., A Carathéodory theorem for the bidisk via Hilbert space methods, Math. Ann. 352(3) (2012), 581–624.Google Scholar
5.Agler, J., Tully-Doyle, R. and Young, N. J., Boundary behavior of analytic functions of two variables via generalized models, Indag. Math. 23(4) (2012), 995–1027.CrossRefGoogle Scholar
6.Agler, J., Tully-Doyle, R. and Young, N. J., On Nevanlinna representations in two variables, J. Funct. Anal. 270(8) (2016), 3000–3046.Google Scholar
7.Bolotnikov, V. and Kheifets, A., A higher order analogue of the CarathodoryJulia theorem, J. Funct. Anal. 237(1) (2006), 350–371.CrossRefGoogle Scholar
8.Bolotnikov, V. and Kheifets, A., The higher order Carathodory–Julia theorem and related boundary interpolation problems, In Recent advances in matrix and operator theory (ed. Ball, J. A., Eidelman, Y., Helton, J. W., Olshevsky, V. and Rovnyak, J.), Operator Theory: Advances and Applications, Volume 179, pp. 63–102 (Birkhuser, Basel, 2008).Google Scholar
9.Carathéodory, C., Über die Winkelderivierten von beschraänkten analytischen Funktionen, Sitzunber. Preuss. Akad. Wiss. (1929), 39–52.Google Scholar
10.Jafari, F., Angular derivatives in polydisks, Indian J. Math. 35 (1993), 197–212.Google Scholar
11.Julia, G., Extension nouvelle d'un lemme de Schwarz, Acta Math. 42 (1920) 349–355.Google Scholar
12.Nevanlinna, R., Asymptotische Entwicklungen beschränkter Funktionen und das Stieltjessche Momentproblem, Ann. Acad. Sci. Fenn. Ser. A 18 (1922), 1–53.Google Scholar
13.Parrott, S., Unitary dilations for commuting contractions, Pacific Math. J. 34 (1970) 481–490.Google Scholar
14.Varopoulos, N. Th., Ensembles pics et ensembles d'interpolation pour les algèbres uniformes, C.R. Acad. Sci. Paris, Sér. A 272 (1971), 866–867.Google Scholar
15.Wlodarczyk, K., Julia's lemma and Wolff's theorem for J*-algebras, Proc. Amer. Math. Soc. 99 (1987), 472–476.Google Scholar